The present invention relates to a design method for a non-rotationally symmetrical optical curved surface such as a progressive power lens or an fxcex8 lens.
In general, an aspherical surface, particularly a complex aspherical surface, is designed after repeated trial and error. An optical performance of a default shape is evaluated, then at least one parameter of the shape is changed and the performance is evaluated. The change of the parameter and the evaluation are repeated until a satisfactory result is obtained.
An aspherical surface can be represented by a mathematical function. In a first conventional method, the entire area of the aspherical surface is represented by a high-order polynomial. A second conventional method divides the aspherical surface into a plurality of areas and a low-order polynomial is defined for each of the areas. In the first method, since the change of the parameter to improve performance of the specific area influences the performance of the entire area, the first method increases number of times of the trial and error to optimize the entire area with considering the influence.
In the second method, since the change of the parameter of the specific area has no influence with the other area, it eases to improve the local optical performance and it is suited for designing the complex aspherical surface. A lens design method using the second method is disclosed in, for example, Japanese Provisional Patent Publication No. Sho 55-146412, or International Patent Re-publication WO 96/11421.
Sho 55-146412 discloses a design method for an aspherical surface of a progressive power spectacle lens. The disclosed method divides the entire area of the aspherical surface into rectangular areas that are divisions divided by a lattice. Each of the rectangular areas is represented by a bicubic polynomial function and coefficients (parameters) of the function are found to solve simultaneous equations so that the values of the expression itself, the differential and the quadratic differential are continuous, respectively. Namely, since the disclosed method aims to provide distribution of the prismatic powers (inclinations of the lens surface), default distribution of the prismatic powers is applied at the beginning, the shape of the lens surface is calculated based on the default distribution, and then the surface shape is represented by the bicubic polynomial functions for the rectangular areas to evaluate the optical performance.
A design method disclosed in WO 96/11421 sets default values of radius of curvature on principal points, and it divides the entire area of the lens surface into rectangular areas that are divided by lattice. Ray tracing is applied to each of the rectangular areas to evaluate the optical performance of the lens, and then the radii of curvature are corrected according to the evaluation. Bi-cubic expressions are defined for the rectangular areas, and then the optical performance of the surface shape is checked. When there is room for improvement in the optical performance, the radius of curvatures are reset to re-define the bicubic polynomial functions. The cycle of check and reset is repeated until the optimum shape is obtained.
However, Sho 55-146412 only discloses the method for evaluating the optical performance of the lens surface, while it does not disclose how to change the parameters. Further, since the method finds the bicubic polynomial functions based on the differentials and the like after setting the refractive prism powers, the prismatic powers must be the parameters of the design even if the surface shape is changed according to the evaluation. However, since the lens shape represented by the distribution of the prismatic powers is not directly related to refractive power and astigmatism that are important items to evaluate the progressive power lens, it is difficult to determine how to change the prismatic powers based on the evaluation to correct aberrations, which increases number of times of the trial and error for the optimization, requiring huge calculations.
On the other hand, since the optical performance can be evaluated by the computerized ray tracing only after the lens shape is represented by the mathematical function, the method disclosed in WO 96/11421 does not actually evaluate the optical performance. Further, the radius of curvature in an X direction must be different from that in a Y direction in the non-rotationally symmetrical surface such as the progressive power lens, while the disclosed method does not distinguish the difference. Moreover, the publication does not show how to determine the lens shape based on the radii of curvature given for the principal points on the lens surface.
It is therefore an object of the present invention to provide a design method of an optical curved surface on the condition that areas of a lens surface divided by a lattice are represented by mathematical functions, which is capable of efficiently calculating a shape of the entire lens surface and a shape of the each area of the lens surface based on curvatures (refractive powers) of the lens surface as parameters that are directly related to aberrations.
For the above object, according to the present invention, there is provided a design method of an optical curved surface, which includes:
dividing an optical curved surface into a plurality of rectangular areas that are divided by the lattice;
defining an original lattice point on the lattice, a backbone line that crosses the original lattice point on the lattice, and standard lattice points that are the lattice points located on the backbone line except the original lattice point;
applying curvatures to all of the lattice points;
applying inclinations to the original lattice point and the standard lattice points;
applying sag to the original lattice point;
calculating a sectional shape of the curved surface along the backbone line based on the sag and inclination of the original lattice point and the curvatures of the lattice points on the backbone line;
calculating sags of the standard lattice points based on the calculated sectional shape;
calculating sectional shapes along orthogonal lines that are orthogonal to the backbone line based on the calculated sags and the applied inclinations of the standard lattice points and the applied curvatures of the lattice points on the orthogonal lines; and
representing the rectangular areas as mathematical functions respectively based on the calculated sectional shapes.
With this method, the sectional shape along the backbone line is calculated with integration based on parameters such as curvatures applied to the lattice points at first, and the sectional shapes along the orthogonal lines are calculated with integration. The surface shape in each of the rectangular areas are, for example, represented by a bicubic polynomial function, which enables evaluation of the optical performance. The curvatures applied as parameters can be replaced with sectional surface powers.
When the parameters are changed based on the evaluation result, the curvature is preferable as the parameter because of its direct relationship with the aberration.
The design method may further include:
evaluating optical performance of the optical curved surface represented by the mathematical functions; and
changing at least one of the applied sag, inclinations and curvatures based on the evaluated result.
The calculation of the sectional shape along the backbone line may include:
a first integration of the curvatures with the value corresponding to the inclination of the original lattice point as an integration constant to obtain a distribution of the values corresponding to the inclinations along the backbone line; and
a second integration of the distribution of the values corresponding to the inclinations with the sags of the original lattice point as an integration constant to obtain the sectional shape along the backbone line.
The calculation of the sectional shape along the orthogonal line may include:
a first integration of the curvatures with the value corresponding to the inclination of the standard lattice point on the orthogonal lattice point as an integration constant to obtain a distribution of the values corresponding to the inclinations along the orthogonal line; and
a second integration of the distribution of the values corresponding to the inclinations with the sags of the standard lattice point on the orthogonal line as an integration constant to obtain the sectional shape along the orthogonal line.
When the optical curved surface has a distribution of the variation of curvature, it is preferable that the rectangular areas are relatively small where the variation of curvature is relatively large and the rectangular areas are relatively large where the variation of curvature is relatively small.
Further, the optical curved surface may be divided by a first lattice whose pitch is relatively large and by a second lattice whose pitch is relatively small. In such a case, the parameters are applied to the lattice points of the first lattice, and the design method further includes:
calculating the parameters for the lattice points of the second lattice by interpolating the parameters applied to the lattice points of the first lattice,
calculating a sectional shape of the curved surface along the backbone line based on the parameters of the lattice points of the second lattice, and
calculating a sectional shape of the curved surface along orthogonal lines that are orthogonal to the backbone line based on the parameters of the lattice points of the second lattice.